How can I remove shading on a plane but still allow shadows from other objects to be cast on to the pane?
The simplest solution I can think of is to use the default Principled BSDF setup (using base color, not emission) and change the type of the light source to Sun. This way, neither cube position, nor Plane position, will change the colors on the plane (other than the colors of areas that go in or exit the shadow).

The reason your examples don't work is that in one case the image is the light, so it can't be shadowed, and in the other you're using a light type with a falloff - simulating the fact that the closer a pixel is to the light source, the more photons from the light source hit that pixel - for the same reason the Sun heats ground the most at noon on Summer. You can lit a surface equally, if it's a fragment of a sphere, with the light source in the center of the sphere - same logic as with curved PC monitors. As you increase the sphere, given the fragment's surface remains constant, it will be closer and closer to a flat surface as it gets further and further from the sphere's center:

If you want a perfectly flat surface, you need an infinitely far light source, and that's what the Sun light type represents.